Compact Totally Real Minimal Submanifolds in a Bochner-Kaehler Manifold

Year 2018, Volume: 1 Issue: 4, 254 - 257, 20.12.2018

Zühal Küçükarslan Yüzbaşı , Mehmet Bektaş Münevver Yıldırım Yılmaz

https://doi.org/10.32323/ujma.422271

Cited By: 1

Abstract

In this paper, we establish the following results: Let $M$ be an $% m-$dimensional compact totally real minimal submanifold immersed in a locally symmetric Bochner-Kaehler manifold $\tilde{M}$ with Ricci curvature bounded from below. Then either $M$ is a totally geodesic or \begin{equation*} \inf r\leq \frac{1}{2}\left( \frac{1}{2}m\left( m-1\right) \tilde{k}-\frac{1% }{3}\left( m+1\right) \tilde{c}\right), \end{equation*}% where $r$ is the scalar curvature of $M.$

Keywords

Bochner-Kaehler manifold, Ricci curvature

References

• [1] B. Y. Chen, Some topological obstructions to Bochner-Kaehler metrics and their applications, J. Dif. Geom., 13 (1978), 547-558.
• [2] B. Y. Chen, K. Yano, Manifolds with vanishing Weyl or Bochner curvature tensor, J. Math. Soc. Japan, 27(1975), 106-112.
• [3] B. Y. Chen, F. Dilen, Totally real bisectional curvature, Bochner-Kaehler and Einstein-Kaehler manifolds, Differential Geom. Appl., 10(1999), 145-154.
• [4] S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math., 28 (1975), 201-228.
• [5] C. S. Houh, Totally Real submanifolds in a Bochner-Kaehler Manifold, Tensor, 32 (1978), 293-296.
• [6] K. Iwasaki, N. Ogitsu, On the mean curvature for Anti-holomorphic p-plane in Kaehlerian spaces, Tohoku Math. J., 27 (1975), 313-317.
• [7] M. Bektas¸, Totally real submanifolds in a quaternion space form, Czech. Math. J., 54(129) (2004), 341-346.
• [8] H. Sun, Totally real pseudo-umblical submanifolds of quaternion space form, Glasgow Math. J., 40 (1998), 109-115.
• [9] L. Ximin, Totally real submanifolds in a complex projective space, Int. J. Math. Math. Sci., 22(1) (1999), 205-208.